| L(s) = 1 | + (1 − 1.41i)3-s − 2i·5-s + (−1.00 − 2.82i)9-s + 4.24·11-s − 1.41·13-s + (−2.82 − 2i)15-s + 2i·17-s + 4.24·23-s + 25-s + (−5.00 − 1.41i)27-s − 8.48i·29-s − 8.48i·31-s + (4.24 − 6i)33-s + 6·37-s + (−1.41 + 2.00i)39-s + ⋯ |
| L(s) = 1 | + (0.577 − 0.816i)3-s − 0.894i·5-s + (−0.333 − 0.942i)9-s + 1.27·11-s − 0.392·13-s + (−0.730 − 0.516i)15-s + 0.485i·17-s + 0.884·23-s + 0.200·25-s + (−0.962 − 0.272i)27-s − 1.57i·29-s − 1.52i·31-s + (0.738 − 1.04i)33-s + 0.986·37-s + (−0.226 + 0.320i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.183953561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.183953561\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 + 1.41i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 + 8.48iT - 29T^{2} \) |
| 31 | \( 1 + 8.48iT - 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 9.89T + 73T^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 2iT - 89T^{2} \) |
| 97 | \( 1 + 7.07T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.608129836183338397697974338284, −8.082181503694393753995430021309, −7.23374834640029010136406053226, −6.43602734136015792264229121213, −5.76543016439949624170954802746, −4.55982760991144897946025918025, −3.87205174948274294178528533749, −2.71167593016259808640293614249, −1.65059741042856963922540564516, −0.71922263061413141927764647362,
1.55372952903544133833840960395, 2.96512269354220909518905118760, 3.27301797183950151530199763237, 4.44210978553710713871516798925, 5.07938130134522124117021659035, 6.26727172036099124958891431267, 6.99672379001929978376647914369, 7.64615540333742443863205655977, 8.803497237253262829011041160317, 9.155827617989315797394631460098
